The last time I saw my grandmother, she was in a hospital bed, somewhere between sleep and unconsciousness, and so mortally still and alabaster white that it was difficult not to conclude that she wasn't already dead.

In many ways, she had spent the past three decades preparing for this eventuality. Each time one of my visits to my grandparents' home in Milan came to an end, both were convinced we would never see each other again, that death would claim them in the interim. And so each successive goodbye became ever more comically cinematic, tears and hugs in lieu of all the things they would never, could never, quite put into words.

But they clung on to the passing of the years with such tenacity that, as I grew out of my 20s and then my 30s, I became increasingly convinced they'd go on to outlive me. My grandfather eventually passed away three years ago, at 94. And then, at 92, after a fall at home left her hospitalised, it seemed that my grandmother's time was at last imminent.

It took me a month to find out about the accident. She hadn't wanted to trouble me, she said later. I flew to Italy, rushed to the hospital and found a very different woman from the one I knew and loved. She was confused and distant, her words slurred. She had always seemed so much younger than her years. Not any more. So when I solemnly kissed her cold forehead on my last day, I learned that all those years of preparation didn't make it any easier to say goodbye. I had not, however, reckoned on that tenacious spirit. Several seasons have since passed and she has recovered; returning home to the daily care of Suzy, her saintly home help, and at least a semblance of her previous life. This week, she turns 93.

My grandparents were always something of a mystery to me. Perhaps grandparents invariably are. They were already old when I was born, and by the time I was a teenager, both had retired, he from the factory floor, she from occasional nursing work. As a child, I would visit every summer; as an adult, I managed about three long weekends a year, each time struck anew by how quiet, how small, their lives were, every day a facsimile of the one before, a trip to the supermarket a bona fide event. Until his contemporaries died off, my grandfather would pass his time playing bowls. But after successive funerals, he rarely ventured out; mornings were spent poring over perpetually incomplete crosswords, afternoons and evenings dozing in front of the television. Truly a man of his era, he relied on his wife for everything, and conveyed his love for her through bickering alone. (He behaved much the same way with me, and I only ever got a hint of his affection via the firmness of the handshakes that welcomed my arrivals, and marked my departures.)

My grandmother's focus, meanwhile, was the upkeep of their small rented flat: beds made, bathroom cleaned, floors polished. The kitchen was her main domain and, deaf to my cries that, actually, a sandwich for lunch would do, she'd impose on me yet more prosciutto, pasta, an extra slice of beef, a little cheese, cake, grapes, espresso. She would take mortal offence if I ever told her enough, stop, I'm full up. As far as she was concerned, I was always too thin and wanted fattening up.

At Milan's main airport, once, my hand luggage was searched before boarding the plane home. Between my clothes, she had hidden dozens of sachets of coffee, packets of saffron, ravioli, a bar of dark chocolate and, for the flight, two ham rolls filled with tomatoes, a lot of butter and a little mayonnaise.

The customs officer looked up at me.

"Nonna," I said in explanation.

To me, my grandparents seemed wonderfully uncomplicated: always welcoming, always there. But that was only ever an artful illusion maintained by their enigmatic silences, their distinct lack of need to fill in their historical gaps for me, to answer fully any questions I ever posed. Both were born during the first world war and came of age in the second. In 1943, my grandmother, born in Yugoslavia, caused a scandal when she got pregnant by an Italian soldier in circumstances she chose never to fully disclose. She fled Slovenia, and, seven mysterious years later, re-established herself in Milan, where she married my grandfather, who agreed to adopt my mother on condition that no one ever learned her true parentage. My mother told me, in confidence, when I was 15, while my grandfather only confessed, nervously, at the age of 89.

They sent my mother to be educated at a nearby convent. She loathed the nuns, and their religion, and wasn't much more enamoured by home life with her overly strict and censorious parents. She left Italy as soon as she could, arriving in London in her early 20s, to marriage, two sons and a divorce my grandfather always maintained he saw coming.

Their relationship remained a difficult one to the bitter end. When she was dying of cancer in 1999, I hoped my mother would seek a final peace with them. But when my grandmother came to visit her in a London hospice, my mother refused to talk, sending the poor woman back home heartbroken. A few weeks later, she died on my grandmother's 80th birthday.

When my grandfather followed her 10 years later, I was working abroad. Because they seem to be in such a hurry to bury people in Italy, I missed the funeral, arriving days later for the unveiling of the headstone. His widow stood by my side, silent and resolute, but I feared this would be the undoing of her, that without him her world would collapse.

I suggested a day out, that we visit an old friend on the other side of Milan. It was wonderful to be out of that claustrophobic flat and in the open world with her on my arm, but she was all too clearly bemused by, and fearful of, the city she no longer knew. When she took a tumble on the escalator in the underground station – I caught her, someone slammed on the alarm, medics were called – she couldn't get back to the sanctity of her flat quick enough, nor to a life whose parameters she alone could set.

My two daughters have only visited their great-grandmother a handful of times. It's always worth the trip to see her face light up when she hugs the girls her own daughter never lived to see, but they are ultimately too loud for her, too chaotic. Milk is spilled, plates broken.

Their relationship is now based on the exchange of photographs and the occasional shy "Ciao" on the telephone. It's not enough, of course, but I'm grateful we have at least that.

It's often suggested that we have better – or less complicated – relationships with our grandparents than we do with our parents. This is certainly true in my case. I cannot hold against them things they did before I was born, and I have only ever known them to be kind and doting, and eminently lovable. There is much I wish I could say to my grandmother now, and much I wish she felt she could say to me. For what it's worth, I offer her all the forgiveness she so clearly craved from my mother.